The high-school part of [ed. American NCTM 1989, I think is being referred to]“standards” contains a list of topics to increase attention, where the first place is given to “the use of real-world problems to motivate and apply theory” (p. 126). What is a “real-world problem”?

Browsing through “standards”, I found quite a few statements about these mysterious critters. On p. 76 (middle-school part) it is said:

“The nonroutine problem situations envisioned in these standards are much broader in scope and substance than isolated puzzle problems. They are also very different from traditional word problems, which provide contexts for using particular formulas or algorithms but do not offer opportunities for true problem solving.”

What? What did they say about traditional word problems? What a nonsence! With their narrow experience the authors pretend to set standards! Are they aware of the rich resourses of excellent traditional word problems around the world? Let us read further:

“Real-world problems are not ready-made exercises with easily processed procedures and numbers. Situations that allow students to experience problems with “messy” numbers or too much or not enough informations or that have multiple solutions, each with different consequences, will better prepare them to solve problems they are likely to encounter in their daily lives”.

Pay attention that the author uses future tense. This means that he or she has never actually used such problems in teaching and never observed influence of this usage on his or her students’ daily lives. He or she has not even invented such problems because he or she does not present any of them. Nevertheless, he or she is quite sure that these hypothetized problems will benefit students. What a self-assurance!

After such a pompous promise it would be very appropriate to give several examples of these magic problems. Indeed, we find a problem on the same page, just below the quoted statement. Here it is:

Problem 48: Maria used her calculator to explore this problem: Select five digits to form a two-digit and a three-digit number so that their product is the largest possible. Then find the arrangement that gives the smallest product.

This is a good problem, although rather difficult for regular school because having guessed the answer, Maria needs to prove it. But the author never mentions the necessity of proof. What does the author expect of calculator’s usage here? It can help to do the multiplications, but it cannot help to prove. It seems that the author expects Maria to try several cases, to choose that one which provides the greatest product and to declare that it is the answer. But what if the right choice never happened to come to her mind? This is very bad pedagogics. Also let us notice that Maria is expected only to “explore” this problem rather than to solve it. According to my vision, exploration is the first stage towards a complete solution. Do the authors expect Maria ever to attain a complete solution? Do they want children to solve problems or just to tamper for a while?

But let us return to our main concern: so-called “real-world problems”. Notice that this problem has none of the qualities attributed to these mysterious critters on the same page: there is neither too much nor not enough information and there are no multiple solutions, each with different consequences.

One colleague noticed that the book still contains some problems described on page 76. Indeed, there are, but in another document. Here is one of them:

Problem 49: You have 10 items to purchase at a grocery store. Six people are waiting in the express lane (10 items or fewer). Lane 1 has one person waiting, and lane 3 has two people waiting. The other lanes are closed. What check-out line should you join?

I have never read any report about usage of this problem. Also I have never read any solution of this problem. Irresponsibility again!

What about problems with too much or not enough informations, they attract much attention in Europe lately, but European scolars want children to treat them critically and in many cases to refuse to solve them! Take for example that famous problem, after which Stella Baruk named her book [Baruk]. In the late seventies, the following problem was given to 97 second and third graders of primary school in France:

Problem 50: There are 26 sheep and 10 goats on a ship. How old is the captain? [Baruk], p. 25

76 children (out of 97) presented a numerical answer obtained by tampering with the given numbers. For instance, they might add the numbers and declare that the captain was 36 years old. Educators of several European countries (France, Germany, Switzerland, Poland) are very preoccupied by the fact that children “solve” unsolvable problems. The European educators would be very pleased if children refused to solve such problems with a comment like “It cannot be solved”. The European educators are quite right. But the same is true of what the “Standards” call “real-world problems”.

The most sound reaction to the problem 49 is “I don’t know”. But what a grade will an American student get after that?

## Friday, December 11, 2009

### From Russia with love: real wor[l]d problems

A friend of mine discovered this paper, Word Problems in Russia and America, by Andrei Toom. Quite a find. A long read, but worth it if you need any moral support in your personal fight against fuzzy math. Here is an excerpt:

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## 12 comments:

Oh my goodness, this is horrifying.

I gave the sheep and goats problem to my 2nd grade son (who scores very high, like 99.9th percentile, on math achievement tests) and lo and behold--he said the captain was 36--no hesitation.

Aaaack!

"Real-world problems" may be those tasks that some teachers identify in the usual bass-ackwards way: "Oh, look at this activity! Why, it teaches fractions and spelling and critical thinking and reflection and... and.... I'm going to use it today with my 4th period groups, because it'll be very motivating! That's what they need. Once they really get into it, they'll learn."

Ooops, sorry, Vicky, my cynicism is showing.

Somewhat off topic, I know but the mention of critical thinking brings to mind one of my pet peeves:

It comes of teaching freshmen, I know, but I hate the term "critical thinking", which, when used by ed students, appears to mean "it's really interesting" or "you have to think about it" but what it really means is "I'm too lazy to think critically enough to be precise about what it is that the problem asks children to do, so I'm just going to call it critical thinking."

Is there now a length limit on comments?

Blogger has always had a length limit on comments -- something like 4300 characters (including spaces). One way to manage this limit successfully is to compose your comment in Word (which has a word/character count feature) and break it into two parts as needed.

I'm nowhere near 4300 characters. Did they lower the limit? Maybe something else is going on. It doesn't give me an error message, it just doesn't post it. Maybe it doesn't like what I'm saying.

This is all I find with a quick check.

"Size of Posts: Individual posts do not have a specific size limit, but very large posts may run you up against the page size limit. (See the next item.)"

I figured it out. I was using a "<" sign, but it seems that I did not get a "tag" error. Oh well. Here goes.

Great find, Vicky!

"With their narrow experience the authors pretend to set standards!"

They want to change math into an image of themsevles. As mentioned in another thread, they see how Kumon works, but they don't believe in it. They see Singapore Math and can't figure out how to trash it. Most use faint praise. I've never quite understood the mentality that allows this to happen. All I can think is that they know only what they know. If you take that away, they have nothing.

"Do they want children to solve problems or just to tamper for a while?"

It's all a cover for lower expectations and full inclusion. The brighter kids get to take the problem further. The school can then apply differential grading. Unfortunately, this self selection process does not help the kids who need it most.

For problems with no one correct answer, the process is not just to apply some sort of Zen-Math thing. There are three typical kinds of problems: m=n, m is less than n, and m is greater than n. In all cases, you start by defining variables and equations. For cases where you have more unknowns than equations, you can define a merit function and learn about search techniques for finding local and global minima. They can study how genetic algorithms work using a different approach. They can start with a bi-section search and learn all about solution bounds, speed of searching, and error bounds. This is real mathematical guess and check.

Another technique is to figure out extra equations (boundary conditions, perhaps) that give you m=n. These are all specific math tools that try to make math un-Zen-like. Isn't that the whole purpose of math? What they are teaching is Anti-Math.

For the case whith more equations than variables, you can apply statistical or least squares techniques. This is what math is all about. Tools and how to apply them. Critical thinking is not something floating out in the ozone all by itself.

Other types of problems might use Monte Carlo methods to solve problems using a simulation technique. For the checkout line problem, that's what they should be studying. Students could begin to examine what math and solution methods go into a Global Climate Model. That's real world. They don't care about rigorous real world problems, they want cover for lower expectations.

I could go on and on. This all quite silly, but why has it gotten this far? You would think that with costs per student in the range of $20,000, that a good solution would appear. The gatekeepers want to protect their turf.

"Problem 48: Select five digits to form a two-digit and a three-digit number so that their product is the largest possible. Then find the arrangement that gives the smallest product."

"... there is neither too much nor not enough information and there are no multiple solutions, each with different consequences."

Incorrect; the problem is badly underspecified, though not in the ways that many teachers would recognize.

So:

1) Do I have to use the same digits in both parts of the problem? "find the arrangement" could mean that I have to use the same digits or it could be that I need to use the arrangement of any five digits that results in the lowest product.

2) I suspect that the writer of the problem intended the student to use five

differentdigits, but this is nowhere specified. The result is that "999, 99" gives the maximum and is trivially easy to find. If the answer to 1) is that I have to use the same digits, the answer there is the same. If I do not, the answer might be "100, 10".3) Unless, of course, the second part of the problem does not require whole numbers. Then, the answer would be ".001, .01"

4) And what about exponents? 9^9^9 is a number represented with three digits, after all.

5) Where is it specified that the answer be in base-10?

6) Leading zeros are conventionally not shown, but there is no requirement that this be so.

It isn't so much that the problem lacks information, though, it's that it was written by somebody not fluent in mathematics. This is qualitatively different than the sheep, goats, and age problem. There are conventional answers to most of the questions above, and in most contexts I wouldn't fault a student for choosing them without explicit specification. But the question of whether reuse of digits (within or between) answers are significant. Any of the available choices could be a reasonable inference from the way the problem is written.

Hmm, my not very mathy kid (8yo) figured out you couldn't find the captain's age but he's not very good nor does he enjoy math. Could it be a complete lack of common sense on the part of the kids? A life that requires very little reasoning from them so there is not 'training' in problem solving?

Here you go, the answer to problem #49:

The Art of Beating Long Lines

And in video form: Good Morning America

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